Permutations with Extremal number of Fixed Points

TL;DR

This paper generalizes the study of permutations with extremal fixed points by considering prescribed descent sets and excedance-based generating polynomials, employing combinatorial and symmetric function techniques.

Contribution

It introduces new permutation statistics and extends existing results to broader classes of permutations with explicit formulas for maximal cases.

Findings

Derived explicit formulas for extremal permutations with prescribed descent sets.

Developed new permutation statistics 'DEZ' and 'lec' with analytical properties.

Applied symmetric function tools to analyze permutation generating polynomials.

Abstract

We extend Stanley's work on alternating permutations with extremal number of fixed points in two directions: first, alternating permutations are replaced by permutations with a prescribed descent set; second, instead of simply counting permutations we study their generating polynomials by number of excedances. Several techniques are used: Desarmenien's desarrangement combinatorics, Gessel's hook-factorization and the analytical properties of two new permutation statistics "DEZ" and "lec". Explicit formulas for the maximal case are derived by using symmetric function tools.